21.4 Exercises

Exercise 21.1 Re-run the moose population model with probabilities of adjusting to the deep snowpack at \(p = 0, \; 0.1, \; 0.9, \;1\). How does that adjusting the probability affect the moose population after 10 years?

 

Exercise 21.2 Modify the two stage moose population model (Equation (21.2)) with the following parameters and plot the resulting adult and juvenile populations:

  1. \(f = 0.6\), \(g = 0.6\), \(p = 0.7\)
  2. \(f = 0.5\), \(g = 0.6\), \(p = 0.4\)
  3. \(f = 0.3\), \(g = 0.6\), \(p = 0.5\)
 
Exercise 21.3 (Inspired by Logan and Wolesensky (2009)) An animal reproduces two, one, or no offspring. The chance it produces one offspring is 0.50, two offspring 0.25, and no offspring 0.25. This animal does not survive after reproducing. Use the function population in the demodeler library to produce 1000 realizations of this stochastic process over 20 generations. Assume the initial population size is 10 individuals. Comment on the long term dynamics of the population.

 

Exercise 21.4 (Inspired by Logan and Wolesensky (2009)) You are playing a casino game. If you win the game earn a dollar. If you lose the game you lose one dollar. The probability of winning or losing is 50-50 (0.50). You start the game with $100. You play the game 200 times. Use the function casino in the demodelr library to produce 1000 realizations of this stochastic process. Comment on the long term dynamics of your earnings. Then assuming the house win probability is 0.52 figure out how long the game, on average, runs before you are broke. Finally, adjusting the house win probability, hypothesize a function of the length of game as a function of the house win probability.

 

Exercise 21.5 Modify the two stage moose population model (Equation (21.2)) to account for years with large snowdepths. In normal years, \(f=0.5\), \(g=0.6\), \(p=0.7\). However for snowy years, \(f=0.3\), \(g=0.6\), \(p=0.5\). Generate code that can account for these variable rates (similar to the moose population model). Plot simulations when \(N=10\) and \(N=30\) and comment on the long-term dynamics of the moose.
 

Exercise 21.6 A population grows according the the growth law \(x_{t+1}=r_{t}x_{t}\).

  1. Determine the general solution to this discrete dynamical system.
  2. Plot a sample growth curve with \(r_{t}=0.86\) and \(r_{t}=1.16\), with \(x_{0}=100\). Show your solution for \(t=50\) generations.
  3. Now consider a model where \(r_{t}=0.86\) with probability 1/2 and \(r_{t}=1.16\) with probability 1/2. Write a function that will predict the population after \(t=50\). Show three or four different realizations of this stochastic process.
 

Exercise 21.7 (Inspired by Logan and Wolesensky (2009)) A “patch” has area \(a\), perimeter \(s\), and a strip (band) of width \(w\) inside the boundary of \(a\) from which the animals disperse. Only those in the strip disperse. Let \(u_{t}\) be the number of animals in \(a\) at any time \(t\). The growth rate of all the animals in \(a\) is \(r\). The rate at which animals disperse from the strip is proportional to the fraction of the animals in the strip, with proportionality constant \(\epsilon\), which is the emigration rate for those in the strip.

  1. Draw a picture of the situation described above.
  2. Explain why the equation that describes the dynamics is \[ u_{t+1}=ru_{t} - \epsilon \left( \frac{w \cdot s}{a} u_{t} \right) \]
  3. Determine conditions on the parameters \(r\), \(w\), \(s\), \(\epsilon\), and \(a\) under which the population is growing.

References

Logan, J. David, and William Wolesensky. 2009. Mathematical Methods in Biology. 1st ed. Hoboken, N.J: Wiley.